New approach for the computation of Flow velocity in partially filled pipes Arranged In parallel

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1 New approach for the computation of Flow velocity in partially filled pipes Arranged In parallel Lotfi ZEGHADNIA, 1, Lakhdar DJEMILI 2, Rezgui Nouredin 1 1 Departement of Civil Engineering, Faculty of Sciences&Technology, University of Mohamed Cherif Messaadia, Souk Ahras, Algeria. Zeghadnia_lotfi@yahoo.fr Tel: Departement of Hydraulics, Faculty of Engineering Science, University of Badji- Mokhtar Annaba, Algeria. l_djemili@hotmail.com 1 Departement of Civil Engineering, Faculty of Sciences&Technology, University of Mohamed Cherif Messaadia, Souk Ahras, Algeria. rezguinordine@yahoo.fr ABSTRACT In this paper, a new approach is presented for the computation of the flow velocity in pipes arranged in parallel based on analytic development. This parameter is needed to trail error procedures to be solved, and have big important in flow measurement and design of drainage network, where the flow is mostly flowing in free surface flow. A new method is elaborated to eliminate the need for trial methods, where the computation of the flow velocity become easy, simple and direct with zero deviation compared to Manning equation and others approaches such as Saatçi(1990) and Akgiray (2005) which have been considered as the best approaches but after investigation these approaches are lack accuracy and do cover the entire range of : Key words: Flow velocity, free surface flow, uniform and steady flow, Circular pipe, Manning equation. Introduction Flow velocity calculation is an important task for hydraulic engineers in the design of conveyance open channels for irrigation, drainage, and sewage network. Sewer and drainage are frequently flowing in free surface flow, in order to avoid the decrease of the flow crosssectional area and thereby the decrease of the flow efficiency. Manning model is already considerate as the best to describe the free surface flow. A number of authors are discussed the model such as (chow 1959),(Henderson 1966),(Metcalf & Eddy, 1981),(Carlier, 1985),,(Hager,2010) and other. Circular shape is the prefer form for sewer system design. Subwatersheds can be arranged in series form or in parallel; similarly the pipes of the sewer and drainage system can be arranged in series or in parallel form. Using Manning equation the computation of the flow velocity and water surface angle is direct and need to trial method

2 and heavy computation. Based on Manning model, a number of researches are trying to propose explicit solution for the free surface flow computation, among the interesting approaches: (Saâtçi, 1990), (Giroud and al, 2000), (Akgray 2004 and 2005) which have trying to eliminate the need to the trial methods, Where the water surface angle range is between 0 and In this study and based on Manning model we will propose a new approach, much simple and more accuracy than the other methods for the computation of the flow velocity partially filled pipes arranged in parallel form, for all the range of surface water angle :, where the maximum deviation is zero. Theory Manning equation: For a long time Manning equation (Manning, 1891), is already considered as the best formula to compute free surface flow due to its simplicity. The Manning equation can be used for uniform flow in pipe. Graphs (camp, 1946), (Swarna, V. and Modak, P., 1990) and tables are established to facilitate the application of Manning equation for the estimation of the flow characteristics (Terence J, 1991). Manning equation can be written as follow: (1) (2) Q: Flow rate in m3/s, : Hydraulic radius, n: Pipe roughness coefficient (Manning n), A: cross sectional flow area, S: slope of pipe bottom, dimensionless, V: flow velocity m/s, To use the Manning model some hypothesis must be assumed: Flow must be steady and uniform, where the slope, cross-sectional flow area, velocity are related to time, and constant in the length of pipe being analysed (Carlier, 1985). The two equations (1) and (2), can be rewritten as function of the water surface angle of the pipe as shown in figure 01 as follow: Figure 01: Water surface angle

3 ( ) [ ( ) ] (3) ( ) [ ( ) ] (4) ( ( )) (05) P (06) ( ( ) ) (07) D : pipe diameter, P : wetted perimeter, : Water surface angle. According to the equations mentioned above, the computation of the flow velocity is direct, which need iterative methods. A number of authors have trying to propose approximate approaches to eliminate the need of trial procedures, where the interesting approaches are Saatçi (Saatçi, 1990), Giroud (Giroud and al, 2000) and Akgiray (Akgiray, 2005). Saatçi approach is based on two steps; first we estimate the water surface angle using the following formula: (08) Secondly, using equation (4), the flow velocity can be calculated. Saatçi equations able to be used for all range of, Where it can only used for range: (Saatçi, 1990). Giroud(Giroud and al, 2000) and Akgiray (Akgiray, 2005) have also propose approaches in order to improve Saatçi equation, where Giroud model is consist to propose direct relationship between average flow velocity and flow rate for between 0 and , where they have estimated the maximum deviation less than 3%. As well, Akgiray have trying to improve more the approaches proposed before, where he have proposed an explicit approximate solutions from Manning equation for four types of problems: 1. Given Q, D, and S, find h/d and (or) V, 2. Given Q, D and V, find h/d and (or) S,

4 3. Given V, D and S, find h/d and (or) Q, 4. Given Q, V and S, find h/d and (or) D. The four problems mentioned above are studied for two cases, when Manning coefficient n is constant, and for n varies with the depth flow as documented by Camp (Camp, 1946), but both are considered as known parameters. In this research, we interest for the first problem, where author (Akgiray, 2005) have propose the following equations to evaluate water surface angle: ( ( ( )) ) (09) (10) The equation (09) is proposed for between 0 and with maximum error equal to 0.72%. The flow velocity can be estimated according to the equations (09) and (04) where the maximum deviation is 0.29%. New approach Flow velocity 1.1.Subwatershed arranged in parallel Subwatersheds may be arranged whether in series form or in parallel, in this study we will focused on the second case, where the pipes are arranged in parallel as shown in figure 02: CC3 CC1 CC2 M 1 M 2 N1 03 N2 Figure 02: Subwatershed and pipes arranged in parallel.

5 The pipe M1-N1 collects water from subwatershed CC1 (which take number 1); The pipe M2-N1 collect water from the equivalent watershed CC2 (which take number 2); The pipe N1-N2 collects water from the equivalent watershed CC3(which take number 3). The flow Q can be evaluating with agree methods, like rational method or SCS method (Viessman and Lewis 2003). Four types of problems are possible to compute the flow velocity and water surface angle according to the reference pipe: 1. The Computation of as function of the pipe 01 characteristics (pipe 01 is the reference pipe). 2. The Computation of as function of the pipe 02 characteristics (pipe 02 is the reference pipe). 3. The Computation of as function of the pipe 01 characteristics (pipe 1 is the reference pipe). 4. The Computation of as function of the pipe 02 characteristics (pipe 2 is the reference pipe). Cases one and two Flow in pipes is steady and uniform, which means the flow characteristics are constant during flow time and in space (for length of pipe being analysed). Let us to consider that pipe M2-N1(or pipe02) is the reference pipe, with known parameters, where the diameter D2, hydraulic radius Rh2, surface water angle, water cross section A2, slope S2, are known data. The slop S3 and roughness n3 are considerate as known parameters for pipes N1-N2 (or pie03). Many Authors considered that Manning equation (Manning 1889) is the best model to describe the free surface flow (Chow 1959), (Saatçi 1990), (Carlier 1985), (Akgray 2004 and 2005), (prabhata and Swamee 1994 and 2004). Equation (04) can be also written as the following forms (zeghadnia and al, 2009): (11) (( ) ( ) ) (12) : Water surface angle in radians as shown in figure 02. is transported in pipe M1-N1 and produced in subwatershed CC1, transported in pipe M2-N1 and produced in subwatershed CC2, is transported in pipe N1-N2 and produced in subwatershed CC3

6 And the ratio Rq 32 between Q 3 and Q 2 and Rq 31 between Q 3 and Q 1, are given by the following equations: (13) (14) (15) (16) (17) (18) From inequations (13) and (14) for just full pipe (under atmospheric pressure) we obtain the following: (19) This is give: (20) Similarly: (21) (22) Based on the equations (15), (17) and (21) the ratio Rq 32 become as: (23) This yield: (24) Using the equation (22), the equation (2) become as follow: ( ) (25) ( ) ( ) (26)

7 From the combination between the equations (24) and (26) give the following: ( ) ( ) ( ) (27) The equation (27) allows the computation of the flow velocity in just full pipe (pipe N1-N2) as function of the reference pipe. In case of partially filled pipe the precedent formula and according to the equation (12) is rewrite as follow: ( ) ( ) ( ) ( ) (28) Similarly, in the case when pipe M1-N1 is the reference pipe, the flow velocity can be calculate as follow: ( ) ( ) ( ) ( ) (29) The equations (28) and (29) give the exact value of flow velocity in pipe N1-N2 as function of the parameters of the reference pipe. Both in the first or in the second case the maximum deviation is zero compared to equation (4). Case three and four In this case, the reference pipe characteristics are known. Here, we will write the characteristics of the first pipe using the second pipe characteristics (reference pipe) as will be shown in the following sections. Using equations (20) and (22), we obtain the following: (30) ( ) (31) Using equation (31) the velocity equation can be written as follow: ( ) ( ) (32) Based on the equations (16), (17) and (19) the ratio Rq 31 become as: This yield: (33) (34) Also, using equations (24) and (34) we get the following equation:

8 (35) Using the equations (32), (35) and (36) we can estimate the velocity using the characteristics of second pipe for just full as follow: (36) ( ) ( ) ( ) (37) For case of partially full pipe, and according to the equation (12), the equation (37) become as: ( ) ( ) ( ) ( ) (38) The equation (38) is the best formula compared to the equation (04) and the recent approaches, where the maximum error is zero as shown in table 01. For case when pipe 01 is the reference pipe, the same result can be obtain as follow: ( ) ( ) ( ) ( ) (39) 1.2.Accuracy Test From The table 01, the equation (38) is an excellent formula, where the maximum deviation is, Compared to Manning equation (04) and approaches of Saatçi (Saatçi, 1990), Giroud (Giroud and al, 2000) and Akgiray (Akgiray, 2005) where the maximum deviation for Saatçi equation is very lack accuracy and for the entire range of. As well the approach of Giroud for between 0 and 360 the maximum error occurs is 12.76%. For Akgiray formula, which is the most recent proposal approach, it is less accuracy too, where for the entire range of the maximum deviation occur is 23.80%. In other hand, the proposal approach in this study using the equation (38) for the first case or the equation (39) for the second case, is very accuracy, where the maximum error occurs is zero, this result is the best until now, where no authors before have claims that a formula can produced the same deviation compared to the equation (4).

9 Table 01: Accuracy test of the equation (38) compared to the recent approaches and equation (4) Manning Equation (04) Proposed Equation (38) Error % Saatçi equation Error % Giroud equation Error % Akgiray equation Error % , E , , E E-06 83, , E , , E , , E , , E E-06 76, , E , , E , , E E-06 65, , E , , E , , E-06 60, , , , , , , , , , ,7916 0, , , , , E-06 44, , , , , , , , , , , , , , E-06 39, , , , , , ,4059 1, , , , , E-06 31, , , , , , E-06 30, , , , , , , , , , , , , , , , ,5811 1,

10 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,416 0, , , , , , , , , , E-06 5, , , , , , ,

11 E E , , , , , , , , , , , Conclusions This paper presents explicit equations for computation of flow velocity in circular sections partially filled based on the reference pipe, where the deviation between these equations and manning equation is zero. It is hoped that the explicit equations will be useful for designers or engineers to appropriately design open channels. References [1] Chow V-T (1959) Open channel hydraulics. Mc Graw-Hill, New York. [2] Henderson,F.M.(1966) OpenChannelFlow,Macmillan,NewYork. [3] Metcalf & Eddy Inc.(1981) wastewater engineering: collection and pumping of wastewater. McGraw-Hill, New York. [4] Carlier, M., (1980) Hydraulique Générale, Eyrolles. [5] Willi H. Hager.(2010). Wastewater hydraulics theory and practice. Second edition, Springer. [6] Saatçi, A.,(1990)Velocity and depth of flow calculations in partially pipes, J. environment Engineering, ASCE., 116(6), pp

12 [7] Giroud, J.P., palmer, B., and Dove, J.E.,(2000) Calculation of flow Velocity in pipes as function of flow rate, J. Geosynthétics International., 7(4-6), pp [8] Akgiray. Ӧ. (2004). Simple formula for velocity, depth of flow and slope calculations in partially filled circular pipes. Environmental engineering science, 21(3), [9] Akgiray. Ӧ. (2005). Explicit solutions of the Manning equation in partially filled circular pipes. Environmental engineering science, 32, [10] Prabhata K.Swamee.(1994). Normal depth equations for irrigation canals. ASCE Journal of hydraulic division,102(5), [11] Prabhata K.Swamee. Pushpa N. Rathie.(2004).Exact solution for normal depth problem. IAHR Journal of hydraulic research,42(5), [12] Achour b., Bedjaoui a. (2006). Discussion of :Explicit Solutions for Normal Depth problem» by Prabhata K. Swamee, Pushpa N. Rathie, IAHR Journal of hydraulic research, 44(5), [13] Manning, R. (1891). On the flow of water in open channels and pipes, Transactions, Institution of Civil Engineers of Ireland, Vol. 20, pp , Dublin. [14] Camp, T.R. (1946). Design of sewers to facilitate flow, J. Sewage Works.,Vol.18., N 01, pp [15] Swarna, V., Modak, P. (1990). Graphs for Hydraulic Design of Sanitary Sewers. J. Environ. Eng.,ASCE., Vol. 116, No. 3, pp [16] Terence.J.Mc (1991). Water Supply and Sewerage, 6 th edition, McGraw Hill, New York. [17] Viessman, W. and G. L. Lewis (2003). Introduction to hydrology (Fifth ed.). Prentice- Hall. [18] ] Zeghadnia, L., Djemili, L., Houichi, L., Rezgui, N., (2009) Détermination de la vitesse et la hauteur normale dans une conduite Partiellement remplie, J. EJSR., 37(4), pp